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Physical Review A

Evaluation of the geometric phase of a two-level atom manipulated on the Bloch sphere using a time-domain atom interferometer

Hiromitsu Imai, Y. Otsubo, Atsuo Morinaga

Published Jul 23, 2007• 9 citations
Publisher

Abstract

The geometric manipulation of the quantum states of two-level atoms with two laser-controlled resonant $\ensuremath{\pi}$ pulses with a relative phase difference was evaluated using a time-domain atom interferometer comprising a cold sodium atom ensemble and two copropagating two-photon Raman pulses. It was verified that the relative phase shift between two states is double the control phase difference and that the rotation around the axis 3 of the Bloch sphere can be controlled by geometric manipulation.

DOI
10.1103/physreva.76.012116

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The geometric manipulation of the quantum states of two-level atoms with two laser-controlled resonant \ensuremathπ\ensuremath{\pi}\ensuremathπ pulses with a relative phase difference was evaluated using a time-domain atom interferometer comprising a cold sodium atom ensemble and two copropagating two-photon Raman pulses. It was verified that the relative phase shift between two states is double the control phase difference and that the rotation around the axis 3 of the Bloch sphere can be controlled by geometric manipulation.
With carefully prepared resonant laser-controlled pulses, the two quantum states of a cold ensemble of sodium atoms were manipulated by pure geometric rotations of the Bloch vector around axes 2 and 3 on the Bloch sphere. The fidelity for two geometric rotations was improved to be 90%. A universal single-qubit operation on the ground-state wave function was demonstrated by a pure geometric operation and the phase shift due to the rotation around axis 3 was measured using a geometric atom interferometer.
The quantum states of cold ensemble of two-level atoms were rotated around axis 2 on the Bloch sphere geometrically using a laser-controlled resonant pulse sequence with pulse areas of $\ensuremath{\pi}∕2$, $\ensuremath{\pi}$, and $\ensuremath{\pi}∕2$, whose central pulse has a relative phase difference of $\ensuremath{\pi}+\ensuremath{\delta}∕2$. It was ascertained that the population probability of the excited state varies according to a sinusoidal function of $\ensuremath{\delta}$ within an uncertainty of 1%. Hybrid operations of two-level atoms were realized combining the geometrical operation with the Rabi excitation around axis 2 or the time evolution around axis 3.
Ramsey spectroscopy and geometric operations on the released sodium Bose–Einstein condensates consisting of 10 7 atoms were demonstrated using copropagating two-photon stimulated Raman pulses, which were resonant to the clock transition between the F =1, m F =0 and F =2, m F =0 states of sodium atoms. The Ramsey fringes with a visibility close to 1 were obtained for two Raman pulses separated by 10 ms. A relative frequency resolution of Ramsey resonance was 6×10 -10 and a phase uncertainty of fringes was 0.06 rad for single measurement. Using resonant laser-controlled Raman pulses, geometric operations for rotations of the two-level Bose–Einstein condensates around axes 2 and 3 on the Bloch sphere were also demonstrated with a relative phase uncertainty of 4%.
We consider the geometrical phases arising in the state vector of two-level atoms due to their interaction with a self-consistently generated classical electrical field propagating without loss through the atomic medium. Three conservation laws are shown to exist generally and are used to solve for the individual quantum amplitudes, phases, and the electric field. We calculate the geometrical phases in two situations: (a) where the atoms are initially in the ground state and (b) where the initial state is a coherent superposition of the ground and excited states. In both cases the geometrical phase is the Aharonov-Anandan phase resulting from the atomic state vector tracing out a closed curve on the projective Hilbert space--here the Bloch sphere. We show that geometric quantities associated with the curve on the Bloch sphere are directly related to physical observables. The solid angle subtended by the closed curve (shown to equal twice the geometric phase) is a measure of the maximum atomic inversion, while the speed with which the curve is traced is related to the energy uncertainty in the state. An experimental method to observe the total phase change in a two-level subsystem is outlined, using photon echoes in a three-level medium.